Question

$\mathrm{what \: is \: value \: of \: k \: if \: remainder \: is \: 30} \\ \mathrm{ \frac{2 {x}^{3} - 7 {x}^{2} + kx + 10 }{x - 5} }$
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Answer 1

Question:

→ What is the value of k if the remainder is 30

(2x³ - 7x² + kx + 10) ÷ (x - 5)

Answer:

k = -11

Step-by-step explanation:

Let:

• f(x) = 2x³ - 7x² + kx + 10
• g(x) = (x - 5)

By Remainder Theorem,

    g(x) = 0

⇒ x - 5 = 0

⇒ x = 5

Now we shall substitute this value of x in the dividend and equate it to the remainder.

f(x) = 2x³ - 7x² + kx + 10

f(5) = 2(5)³ - 7(5)² + k(5) + 10 = 30

⇒ 2(125) - 7(25) + 5k + 10 = 30

⇒ 250 - 175 + 5k + 10 = 30

⇒ 75 + 5k + 10 = 30

⇒ 5k + 85 = 30

⇒ 5k = 30 - 85

⇒ 5k = -55

⇒ k = -11

∴ The value of k = -11

Answer 2

Given :-

• Remainder (R) = 30
• Divisor (D) = x - 5

Find :-

• Value of K

Solution :-

Find the Value of X

Taking Divisor ,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm{x - 5 = 0} \: \: \\ \: \: \: \: \: \: \: \: \: \: \boxed{\rm{x = 5}}$

Now, Put the Value of X in 2x³-7x²+ kx+10=30

$\: \: \: \: \rm{2{(5)}^{3} } - 7 {(5)}^{2} + k(5) + 10 = 30\\ \: \: \: \rm{ 2 \times 125 - 7 \times 25 + 5k + 10 = 30} \\ \rm{250 - 175 + 5k + 10 = 30 } \\ \rm{5k = 30 - 250 + 175 - 10} \\ \rm{5k = 205 - 260} \\ \rm{5k - 55} \\ \rm{k = \frac{ - 55}{5} } \\ \rightarrow\boxed{\green{ \rm{k = - 11}}}$

#Gunjan